In the middle of the 19th century, one of the most widely publicised scientific struggles was to predict the motion of planetary bodies. Using Newtonian mechanics, it was simple enough to calculate the trajectory of one or two planets. However, when a third was added into the mix, the equations become complex and incomprehensible.
Steven Strogatz described the three body problem in a 2009 article:
It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the… equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he turned his attention to the three-body problem for the sun, earth and moon. He couldn’t solve it, and neither could anyone else. It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.
Newton knew nothing about chaotic dynamics, but he did tell his friend Edmund Halley that the three-body problem had “made his head ache, and kept him awake so often, that he would think of it no more.”
Newton wasn’t the only one. Two centuries on, in honour of his 60th birthday, Oscar II, King of Sweden established a prize for anyone who could find the solution to the so-called three-body problem.
The prize was eventually given to French Mathematician Henri Poincaré. Poincaré did not in fact manage to solve the three body problem. But as one of the judges put it ‘while this work cannot indeed be considered as furnishing the complete solution of the question proposed… it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics.’
Poincare found the trajectory of planets in the three body system to be one of ‘awesome complexity’ (Capra, 1996), and in describing it he inadvertently developed one of the first fractals. As he described it:
When we try to represent the figure formed by these [planets] and their infinitely many intersections, [they] form a type of trellis, tissue, or grid with infinitely fine mesh [which] bends back upon itself in a very complex manner… I shall not even try to draw it, [yet] nothing is more suitable for providing us with an idea of the complex nature of the three-body problem.
Poincare was, in the poetic words of Fritjof Capra, ‘gazing at the fingerprints of chaos’. With the advent of computers over a hundred years later, scientists were at last able to represent Poincare’s figures. Here’s an early example, with each of the three colors represents the trajectory of a distinct body:
(for more on this, and exotic-sounding things such phase space and strange attractors, see the ODI working paper on complexity and aid – concept 7)
The comparison between such systems and social, economic and political issues is not a perfect one. But as Deborah Barlow has written, as a catch all for what lives outside our model of reality, the three-body problem is at least a useful metaphor.
Certainly, the turbulence that has been experienced in the past few weeks of British politics, and the way the situation is continuing to unfold, is analogous to a two body problem evolving into a three body problem.
We saw this in the debates, when the surprising popularity of the Lib Dem leader left pundits struggling to assess the outcome of the election – ‘the most unpredictable in a generation’. We saw it on election night, when the distribution of votes confounded all attempts by the exhausted TV presenters to explain what was happening. And we are seeing it now in the negotiations, with each party having to think through a range of possible future permutations and combinations, with each move potentially triggering a dizzying number of consequences. We also have to look forward to the potential complexity of the legislative and policy-making arrangements under a hung parliament with three parties.
Get it right, and it could make for the most interesting and tranformative period in British politics for decades. Get it wrong, and things could descend into turbulence and chaos.
Newton, one imagines, would be chuckling to himself in the grave.